Mavni Program, Which Has Drawn About 3,000 Enlistees, Is Up for Renewal in September

Asim Manizada says he felt lucky last year after he was picked in a U.S. government lottery for an H-1B, a skilled-worker visa that lets him work in the U.S. for at least three years. But Mr. Manizada still faced a long and uncertain path to permanent residency and citizenship.

So in February, the Azerbaijani man initiated the process to join a U.S. military program that offers citizenship within months to individuals skilled in...